## Abstract

Let ξ be a smooth vector bundle over a differentiable manifold M. Let h : ε^{n - i + 1} → ξ be a generic bundle morphism from the trivial bundle of rank n - i + 1 to ξ. We give a geometric construction of the Stiefel-Whitney classes when ξ is a real vector bundle, and of the Chern classes when ξ is a complex vector bundle. Using h we define a differentiable closed manifold over(Z, ̃) (h) and a map φ{symbol} : over(Z, ̃) (h) → M whose image is the singular set of h. The ith characteristic class of ξ is the Poincaré dual of the image, under the homomorphism induced in homology by φ{symbol}, of the fundamental class of the manifold over(Z, ̃) (h). We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.

Original language | English |
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Pages (from-to) | 1220-1235 |

Number of pages | 16 |

Journal | Topology and its Applications |

Volume | 154 |

Issue number | 7 SPEC. ISS. |

DOIs | |

State | Published - 1 Apr 2007 |

### Bibliographical note

Funding Information:* Corresponding author. E-mail addresses: marcelo@matem.unam.mx (M.A. Aguilar), jlcm@matcuer.unam.mx (J.L. Cisneros-Molina), eduardo@gauss.mat.uson.mx (M.E. Frías-Armenta). 1 Partially supported by Proyecto CONACyT G36357-E.

## Keywords

- Characteristic classes
- Generic bundle morphisms